Color Mode:
Nine-Point Circle (in red)
Incircle (in orange)
Exircles (in green)
Triangle ABC (in blue)
Others (in gray )
Point 
: Intersection of incircle and exircles.
Nine special points of an arbitrary triangle ABC, in particular, the feet of the altitudes, the intersections of the medians with the opposite sides, and the midpoints of the segments from the vertices to the orthocenter, lie on a common circle, the so-called nine-point circle of ABC. Another result presently of interest to us is the concurrence of the three angle bisectors of ABC at a point, the incenter, that is the center of a circle, the incircle, tangent internally to each of the sides of ABC. Finally, a related result is that if the sides of ABC are extended, three additional circles, called excircles can be constructed, each tangent externally to the sides of ABC.
A German high-school teacher, K. Feuerbach, had the good fortune of discovering an amazing fact. For any triangle ABC, the nine-point circle is tangent to the incircle and to the three excircles.